There’s a pretty thought experiment that’s sometimes attributed to Democritus though it’s actually due to a later popularizer of the atomic hypothesis1 and it goes like this: Suppose we use the world’s sharpest knife to cut a block of cheese in half, leaving two small blocks where before there was one large one. If cheese is made of atoms, some of the atoms end up in one half and the rest end up in the other. But if cheese is a continuous substance, and the block of cheese is analogous to a line segment in Euclidean geometry, then what happens to the points that are exactly lined up with the knife’s edge? Do they get duplicated? Do they vanish? Does the knife somehow slip to one side or the other of those knife’s-edge points? None of these options seems satisfying, but if we’re cutting something that’s truly continuous, what other options do we have?
In some ways the slipping-knife option, in which the symmetry of the block of cheese gets broken, seems the least satisfying to me. It suggests that you can never truly cut something into two equal pieces, not because of the imprecision of human instruments but because of something inherently strange about space – a strangeness that affects even the bare-bones one-dimensional kind of space that Euclid didn’t find interesting enough to say much about. Yet curiously, the third option in the parable, in which the knife must slip either to the left or to the right, received a kind of vindication a century and a half ago when German mathematician Richard Dedekind set out to think deeply about what real numbers are.
CUTTING THE NUMBER LINE
“How do you define ‘real’?” — Morpheus, “The Matrix”
Richard Dedekind was the last doctoral student of Carl Friedrich Gauss, who is regarded by many as the greatest mathematician of the nineteenth century. We met Dedekind in my essay When Five Isn’t Prime. After receiving his habilitation in 1854, he lectured on probability and geometry at Göttingen before his growing reputation as a teacher enabled him to take on a position at the Polytechnikum in Zürich. At the Polytechnikum he started teaching calculus classes and began to reconsider the foundations of the subject and the nature of real numbers. Although he didn’t publish the resulting essay “Continuity and irrational numbers” until 1872, his ideas were fully formed by the end of 1858.
In elementary school we’re taught to depict numbers as living on a line and this picture becomes so deeply ingrained that it’s easy to forget that the correspondence between numbers and points was once new.2 We learn to plot whole numbers and negative numbers and fractions as individual points on the number line. When the number a is less than the number b, the point corresponding to a on the number line lies to the left of the point corresponding to b.
In middle or high school we learn about new kinds of numbers: irrational numbers that can’t be written as fractions. The rational and irrational numbers taken together constitute the real numbers. We also move beyond plotting individual points to plotting sets of points; for instance we’re taught to represent the set of numbers x satisfying 0 ≤ x ≤ 1 as a segment of length 1 on the number line and to represent the set of numbers x satisfying 0 < x < 1 as an ever-so-slightly different segment of length 1, the difference between the two segments being that the former contains both of its endpoints while the latter contains neither of them. We call the former set of numbers the closed interval with endpoints 0 and 1 and the latter set of numbers the open interval with endpoints 0 and 1. To represent the tiny difference between the two intervals we’re taught to draw them as respectively shown in the image below:
Those empty and filled-in circles aren’t meant to be taken literally; after all, the endpoints of an interval are points, not circles. We’re also taught to represent the closed interval by [0, 1] and the open interval by (0, 1). For that matter, we can also talk about the half-open, half-closed intervals [0, 1) and (0, 1], each of which contains one of its endpoints but not the other.
The entire real line can be written as (−∞,+∞), where −∞ and +∞ are to be understood not as endpoints but as indicators that the set goes on forever — to the left in the case of −∞, to the right in the case of +∞. So for instance the whole real line can be split into the set of negative numbers, written as (−∞, 0), and the set of nonnegative numbers, written as [0, +∞). These sets are an open ray and a closed ray respectively, where a ray is said to be closed if it contains its endpoint and open if it omits its endpoint. Every real number belongs to exactly one of these two sets. Notice that there is a leftmost element of [0,∞), namely 0, but that there is no rightmost element of (−∞, 0); for, if a is an element of (−∞, 0), then a/2 is another element of (−∞, 0) that’s further to the right than a.
You can also split (−∞, +∞) into the closed ray (−∞, 0] and the open ray (0, +∞). Now it’s the ray on the left that’s closed and has a rightmost element and the ray on the right that’s open and has no leftmost element.
When, following in Dedekind’s footsteps, I teach a calculus class that’s aimed at students who really want to understand things, I like to start by asking the students to think of all the different ways you could split the number line into two nonempty pieces A and B, so that every number belongs to exactly one of the two pieces and every number in A is less than every number in B.
After a flurry of exploration, the students conclude that for each real number c they get two ways of splitting (−∞, +∞) into two pieces of the required kind: one with A = (−∞, c) and B = [c, +∞) and one with A = (−∞, c] and B = (c, +∞), as shown below.
The numbers less than c get assigned to A, the numbers greater than c get assigned to B, and c itself can be assigned to either set. In one case, A has a largest element while B has no smallest element; in the other case, it’s the other way round.
But after finding these ways to split the real line, the students get stuck. They try to find more ways to split the real line into a left-set and a right-set, but every new way they find turns out to be a disguised version of a way they’ve already found. After a minute or so of stuckness, students find it easy to accept that there probably aren’t any more ways to do it, and my suggestion “Why don’t we just accept it as an assumption?” is readily endorsed.
The students don’t realize it, but they’ve just swallowed a red pill that among other things will lead them to accept both the existence of irrational numbers and the fact that .999… equals 1 a few weeks in the future.
DOWN THE RABBIT HOLE
“After this, there is no turning back.” — Morpheus, “The Matrix”
The red pill is Dedekind’s completeness axiom, which he stated as follows: “If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.”
To the potential criticism that such a banal assertion seems hardly worth much fanfare, Dedekind wryly remarked “I am glad if everyone finds the above principle so obvious and so in harmony with his own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has anyone the power.”
Note those last five words, in which Dedekind says in effect “I can’t prove it but neither can you.” Why was Dedekind so sure that his inability to derive his completeness axiom from more fundamental geometric principles wasn’t a result of his own lack of cleverness? It’s because the axioms of Euclid are intimately related to the straightedge-and-compass method of construction, which by 1858 was already known to be inadequate to the needs of geometry. For instance, Pierre Wantzel had proved in 1837 that given a line segment of length 1, there’s no way to use a straightedge and compass to construct a line segment whose length is the cube root of 2. Consequently, there’s no way to use Euclid’s axioms to show that there exist two line segments such that the length of the longer segment divided by the length of the shorter segment equals the cube root of 2.3 Or to put it concisely, you can’t use Euclid’s axioms to show that the cube root of 2 exists. On the other hand, as I’ll sketchily argue below, once you accept Dedekind’s axiom, you can prove that the cube root of 2 exists. So Dedekind could know with certainty that his assertion wasn’t a hidden consequence of Euclid’s axioms for geometry.
Of course, mathematicians have never seriously doubted that in some sense the cube root of 2 exists even if they argued about whether it deserved to be called a number. It seems intuitively clear that if the number x = 5/4 satisfies x3 < 2 while the number x = 4/3 satisfies x3 > 2 then surely some number or number-ish thing between 5/4 and 4/3 should satisfy x3 = 2 on the nose. Indeed, a famous nineteenth century theorem called the Intermediate Value Theorem makes this kind of reasoning respectable. It asserts that, since the unbroken curve y = x3 starts from below the line y = 2 and then goes above the line y = 2 as one follows the curve from left to right, there must be a crossing-point where the curve hits the line. It makes visual sense.
But why is the Intermediate Value Theorem true? How can one prove it? Like Dedekind’s completeness axiom, it can’t be derived from the axioms of Euclid. In fact, the Intermediate Value Theorem turns out to be logically equivalent to Dedekind’s completeness axiom; each can be derived from the other.4
There are number systems that don’t satisfy the Intermediate Value Theorem, and if you draw a number line for a number system like that, it won’t satisfy Dedekind’s completeness axiom either. These are the “fakes” in the title of this essay. Dedekind’s completeness property is the feature that distinguishes the real number system from its many impostors.
The simplest impostor mathematicians study is the set of dyadic rationals: numbers that can be written as multiples of some power of 1/2. If you imagine a ruler divided into inches, half-inches, quarter-inches, eighths-of-an-inch, and sixteenths-of-an-inch, and imagine carrying out the bisection procedure to infinity, you’d get the dyadic rationals. It’s not a bad number system, and it’s got a key property that distinguishes it from a super-fake number system like the set of multiples of 1/1,000,000: for any two dyadic rational numbers, there’s a dyadic rational number in between them — infinitely many, in fact. You can approximate any number as closely as you wish using dyadic rational numbers.
A more capacious number system is the set of terminating decimals, that is, numbers that can be written as multiples of some power of 1/10. Again, you can use it to approximate numbers as closely as you like, and as long as you confine yourself to addition, subtraction, and multiplication, you’ll be fine. This number system is good enough for many down-to-earth applications of mathematics. But try to divide one number by another and you’ll often hit a wall; for instance, within this system you can’t divide 1 by 3.
The simplest fake version of the real number system that lets you add, subtract, multiply, and divide with wild abandon is the rational number system. It’s fine for many purposes, but not for doing calculus, and not even that good for doing algebra. If you draw the preceding picture of the curve y = x3 using only points whose x-coordinate and y-coordinate are both rational, you’ll find that the curve goes from below the line y = 2 to above the line y = 2 without ever actually crossing that line!
Although the completeness axiom can be viewed as a negative assertion about ways of splitting the real numbers — “You’ll never find any other way to split the number line into a left piece and a right piece besides the obvious ones” — it can be used in a positive way to prove the existence of specific numbers; for, given a splitting of the number line into two nonempty sets A and B such that every element of A is less than every element of B, the axiom tells us that there’s got to exist a number c such that c is either the largest element of A or the smallest element of B. For instance, let A be the set of all real numbers x satisfying x3 < 2 and let B be the set of all real numbers x satisfying x3 ≥ 2. Dedekind’s axiom says that there must exist a number c such that c is either the largest element of A or the smallest element of B, and from this it can be shown algebraically that c3 is exactly 2.5
As for why Dedekind’s completeness axiom implies that .999… = 1 (an equation I’ve written about here and here), a full explanation would require a detour into what we mean when we write .999…, but here’s a hint into why .999… must equal 1 in any Dedekind-complete number system. If 0.999… and 1.000… were different, the difference between them would be a positive number that’s infinitesimal, i.e., one that’s smaller than all the numbers 1/10, 1/100, 1/1000, etc. Let’s show that this conflicts with Dedekind’s axiom. Let A be the set consisting of all the negative numbers, the number zero, and all the infinitesimal positive numbers, and let B consist of everything else, that is, all the positive numbers that aren’t infinitesimal. It isn’t hard to show that if x is an infinitesimal positive number, then the larger number 10x is infinitesimal too, from which we conclude that A doesn’t contain a largest element. On the other hand, it isn’t hard to show that if x is a positive number that isn’t infinitesimal, then the smaller number x/10 isn’t infinitesimal either. So B doesn’t contain a smallest element. Taking stock of the situation, we see that we’ve split our number system into a left set that has no rightmost element and a right set that has no leftmost element.
Where does that leave us? Not in Dedekind’s real line, that’s for sure!
NO TURNING BACK
“Free your mind.” — Morpheus, “The Matrix”
By accepting Dedekind’s completeness axiom, my calculus students have, without realizing it, jumped over centuries of confusion and arrived at the modern notion of the continuum. Yes, I’ve tricked them into doing it, but I won’t apologize for that. Sooner or later – though, in most colleges, later rather than sooner – math majors get introduced to some version or variant of the completeness axiom for the real numbers, usually in a class with the somewhat opaque name “real analysis”.6 Why not show them a completeness axiom that’s intuitively plausible? Dedekind’s completeness axiom is the most intuitively plausible of all the logically equivalent ways of formalizing the intuition that the real line, unlike the many kinds of fake line, doesn’t have holes in it.
Here’s the way I like to state the completeness axiom: If the number line L is a real line and not a fake one, then there’s no way to divide L into two nonempty sets A and B such that
(1) A and B have no number in common,
(2) A and B together contain every number,
(3) Every number in A is less than every number in B,
(4) A has no largest element, and
(5) B has no smallest element.
That’s it! From this axiom about the real line you can deduce that all the numbers that calculus needs actually exist.
Except then you can start to worry: How do we know that the real line exists?
Of course I don’t mean “exists” in the physical sense. I mean it in the mathematical sense, though the concept of “existence in the mathematical sense” is murky and controversial.
You can certainly take a fake line and make it less fake by creating new numbers to fill its holes.7 Say we start with the fake version of the real numbers consisting of the rational numbers and nothing else. Throw in the square roots of all the positive rationals and you’ll fill some of the holes in the rational number line, but there’ll still be holes associated with more complicated numbers like sqrt(2+sqrt(3)). If you want a hole-free number line you’ll have to throw them in as well, along with all their square roots, and new sums and products, and so on.
Even if, through this process of throwing in sums and differences and products and quotients and square roots of already-included numbers, somehow carried on for infinitely many steps, you were to reach a resting point from which no further progress could be made along those lines, you’d only have arrived at the set of straightedge-and-compass–constructible real numbers. Your line would still have holes in it; for instance, no cube root of two.
So maybe you’d work harder and enlarge your repertoire of operations for what new numbers to throw in, so you could get cube roots and fourth roots and whatnot. Then at the end of that infinitely long day you’d have constructed all the algebraic real numbers, but you’d still be missing important numbers like π. In the spot where π is supposed to be there’d be a hole in the number line; not a gaping hole, by any means, but still a disqualifying defect. And after you fill your π hole, there’d still be an e hole to fill, and …
I could continue this parable of unending enlargement, but I think you see how the story ends, or rather, how it fails to end. You’ll never fill all those holes.
So what entitles you to imagine that somewhere out there is a magical number system whose holes have all been filled?
Dedekind came up with a kind of answer to that, in the form of what are called “Dedekind cuts”. With this one simple device, Dedekind gave a way to create all the irrational numbers at once. And that’s what I’ll write about next month.
This essay is a supplement to chapter 4 (“Holes Too Small to See”) of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds cool and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!
ENDNOTES
#1. I learned about the parable from https://van.physics.illinois.edu/ask/listing/24315 which quotes an extensive passage from Scott Aaronson’s book Quantum Computing Since Democritus that in turn adapts the parable from a paraphrase given by Carl Sagan. Interested readers who want to dig deeper should see the Stackexchange discussion of the knife story.
#2. For more on this, see the Stackexchange discussion of who invented the number line.
#3. Actually, it’s a bit of an anachronism to imagine the Greeks talking about dividing
one length by another; rather, they would have expressed themselves in terms of ratios of lengths. I’ll have more to say about ratios of lengths next months, but for now let’s ignore this nicety.
#4. To learn more about the equivalences between various completeness properties of the real numbers, see my 2012 American Mathematical Monthly article Real analysis in reverse.
#5. The details are messy, but the core idea is simple. If c3 were smaller than 2 then there’d have to be a number c′ slightly bigger than c with the property that (c′)3 is still smaller than c. On the other hand if c3 were bigger than 2 then there’d have to be a number c′ slightly smaller than c with the property that (c′)3 is still bigger than c. Either way we get a contradiction.
#6. In real analysis courses it’s common to assume the least upper bound property, which is logically equivalent to Dedekind’s completeness axiom. The pedagogical problem with the least upper bound property is that it concerns arbitrary sets of real numbers that are bounded from above; in thinking about the least upper bound property, you have to picture an unknown set S of real numbers, where all you’re told about S is that there exists some real number b such that every element of S is less than or equal to b. What does such a set typically look like? I have no idea, and more importantly, neither do my students. It could be fractal or even non-measurable; who knows? In contrast, it’s much easier to picture — or at least to convince yourself that you’re picturing — a generic way of splitting the number line into a left set and a right set. For more on the pedagogy of introducing the real numbers via Dedekind’s completeness axiom, see the slides from my 2010 talk Dedekind’s forgotten axiom and why we should teach it.
#7. If you’re really the worrying kind, you could worry about what entitles you to create numbers just by specifying what properties you want them to have.
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